I'm a video guy, but lets see if I can describe this correctly... If not I'm sure Rich, Scott, Randy, Ryan, Joe, etc.. will jump in and correct me.

FX Lens = Designed for Full Frame sensor. It has more glass, is a heavier lens, and more expensive
DX Lens = Designed for cropped sensor. It has less glass, is lighter, and cheaper.

Using the Nikon 14-24mm lens as an example...

You can use an FX lens on a DX body, but then you have to multiple the focal length by the sensor's crop factor to get the "true" focal length. In your case the "crop factor" is 1.5, so the 14-24mm lens will effectively become a 21-36mm

Now if you try the inverse and put a DX lens on an FX body then you will get something like this: [*]
** The cool thing is that nikon has a "crop mode" on the higher end cameras that will adjust for this, so your picture won't have the black borders showing from the inside of the DX lens.

And finally if you use an FX lens on an FX body then you get a true 14-24mm.

Most DSLR cameras use sensors that are physically smaller than the size of a piece of 35mm film. So since the sensor is smaller, it captures less data...so you wind up with an image that shows the middle of the lens, and extends as far to the left/right/up/down as the size of the sensor will permit. The image that DPC linked to above is a cool example of how much data a DX format (1.5x crop) will capture when compared to what a 35mm film camera would capture (the camera used to take this picture is an FX mount Nikon DSLR). So when you use a DX format camera body, you're essentially losing the outer edges of the photo, though that isn't usually a huge deal until you get into the wide angle and fisheye lenses. That's the quick-n-dirty of it- there's a bunch of small details I left out for simplicity's sake.

As for the lens rental, I would absolutely go with that 14-24mm. I've read some pretty great things about it.